TY - JOUR
T1 - A pressure-robust embedded discontinuous galerkin method for the stokes problem by reconstruction operators
AU - Lederer, Philip L.
AU - Rhebergen, Sander
N1 - Funding Information:
\ast Received by the editors February 12, 2020; accepted for publication (in revised form) July 27, 2020; published electronically October 14, 2020. https://doi.org/10.1137/20M1318389 Funding: The work of the first author was supported by the Austrian Science Fund (FWF) through the research program ``Taming complexity in partial differential systems"" (F65) and the project ``Automated discretization in multiphysics"" (P10). The work of the second author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant RGPIN-05606-2015. \dagger Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria (philip.lederer@ tuwien.ac.at). \ddagger Department of Applied Mathematics, University of Waterloo, Canada ([email protected]).
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics
PY - 2020/10/14
Y1 - 2020/10/14
N2 - The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a pointwise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming, and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right-hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor-Hood and mini elements by Lederer et al. [SIAM J. Numer. Anal., 55 (2017), pp. 1291-1314]. We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods.
AB - The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a pointwise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming, and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right-hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor-Hood and mini elements by Lederer et al. [SIAM J. Numer. Anal., 55 (2017), pp. 1291-1314]. We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods.
KW - Discontinuous Galerkin finite element methods
KW - Embedded
KW - Exact divergence-free velocity reconstruction
KW - Pressure-robustness
KW - Stokes equations
KW - n/a OA procedure
UR - https://www.scopus.com/pages/publications/85095971270
U2 - 10.1137/20M1318389
DO - 10.1137/20M1318389
M3 - Article
SN - 0036-1429
VL - 58
SP - 2915
EP - 2933
JO - SIAM journal on numerical analysis
JF - SIAM journal on numerical analysis
IS - 5
ER -